CNN's vs Densely Connected NN's
The keyword here is Parameter Sharing or Weight Sharing across various image portions.

If we take a simple example of grayscale binary image of an alphabet 'F', it is a combination of multiple patterns. The patterns here are vertical lines and horizontal lines. These patterns are based on relation between intensities of contiguous cells. This relation between contiguous cells is established using a weight matrix.  

Also, for identifying multiple horizontal lines, we dont need multiple node-sets in hidden dense layer trying to identify different horizontal lines in the image. The pattern is same but present in different locations. Hence the sharing of weights comes into picture. 

In the 1st hidden layer, encode the pattern horizontal line in a weight matrix(learnt during training and used in testing). Place it over small grid and check for presence. As this matrix is slided and tested across the image, the presence of horizontal lines is marked in various locations. This weight matrix is called a kernel. 

Combining the above points, kernel provides a way of handling parameter / weight sharing between contiguous cells to identify patterns. Dense layer instead of kernels would solve it eventually but start in a random manner. Since a efficient way was identified, it is being used. 

Next to identify vertical lines, another kernel needed and slide across.

Suppose next we have dense layer as 2nd hidden layer. This layer looks for combination of patterns ('p' horizontal lines and 'q' vertical lines in this case for 'F') present and learns combinations to identify output. 

Just to compare with traditional programming, kernels are like regular expressions. dense layers are like loops. Just sharing my thoughts. Any better explanation is welcome.